Exponents
The idea
An exponent is compressed multiplication: 3⁵ means 3 × 3 × 3 × 3 × 3, five copies of 3 multiplied together, just as multiplication once compressed repeated addition for you. The base tells you what to copy and the exponent tells you how many copies. This shorthand is how math handles things that grow by repeated doubling or tripling — chain messages, folded paper, populations — where amounts explode far faster than addition can describe.
Every exponent rule falls out of counting copies: 3² × 3⁴ is two copies times four copies, six copies in all, so it equals 3⁶ — multiply same-base powers by adding exponents. The number-one trap is reading 3⁵ as 3 × 5 = 15, when it is actually 243; the exponent counts factors, it is not a factor itself. Also keep base and exponent straight: 2³ = 8 while 3² = 9, so the two jobs are not interchangeable.
Worked example
You send a funny video to 3 classmates. The next day, each person who received it that day sends it to 3 new people, and this repeats every day. How many new people receive the video on day 5?
- Track the first days to find the pattern: day 1 reaches 3 people, and day 2 reaches 3 × 3 = 9, since each of the 3 sends to 3 more.
- Each day multiplies the previous day's count by 3, so day n reaches 3^n people; day 5 is 3⁵.
- Build it up one factor at a time: 3² = 9, 3³ = 27, 3⁴ = 81, 3⁵ = 243.
- Contrast with the common misreading: 3 × 5 = 15 would be the count if 3 new people were added each day, but repeated multiplication reaches 243 — exponents grow far faster than addition.
Answer. On day 5 the video reaches 3⁵ = 243 new people.
Check your understanding
- Why does multiplying powers of the same base let you add the exponents — what is being counted?
- How would you explain the difference between 3 × 5 and 3⁵ to someone meeting exponents for the first time?
- What should 3⁰ equal if the pattern of dividing by 3 as the exponent drops is to keep working?
- Where in real life does repeated multiplication beat repeated addition so dramatically, and why does that matter?
Build the foundations first
Exponents builds on these concepts. If any feel shaky, start there.